235. Let us now suppose that the sides of a b c d, instead of being a whole number of inches, contain some inches and a fraction. For example, let a b be 3½ inches, or (114) ⁷/₂ of an inch, and let a c contain 2½ inches, or ⁹/₄ of an inch. Draw a e twice as long as a b, and a f four times as long as a c, and complete the rectangle a e f g. The rest of the figure needs no description. Then, since a e is twice a b, or twice ⁷/₂ inches, it is 7 inches. And since a f is four times a c, or four times ⁹/₄ inches, it is 9 inches. Therefore, the whole rectangle a e f g contains, by (234), 7 × 9 or 63 square inches. But the rectangle a e f g contains 8 rectangles, all of the same figure as a b c d; and therefore a b c d is one-eighth part of a e f g, and contains ⁶³/₈ square inches. But ⁶³/₈ is made by multiplying ⁹/₄ and ⁷/₂ together (118). From this and the last article it appears, that, whether the sides of a rectangle be a whole or a fractional number of inches, the number of square inches in its surface is the product of the numbers of inches in its sides. The square itself is a rectangle whose sides are all equal, and therefore the number of square inches which a square contains is found by multiplying the number of inches in its side by itself. For example, a square whose side is 13 inches in length contains 13 × 13 or 169 square inches.
236. EXERCISES.
What is the content, in square feet and inches, of a room whose sides are 42 ft. 5 inch. and 31 ft. 9 inch.? and supposing the piece from which its carpet is taken to be three quarters of a yard in breadth, what length of it must be cut off?—Answer, The content is 1346 square feet 105 square inches, and the length of carpet required is 598 feet 6⁵/₉ inches.
The sides of a rectangular field are 253 yards and a quarter of a mile; how many acres does it contain?—Answer, 23.
What is the difference between 18 square miles, and a square of 18 miles long, or 18 miles square?—Answer, 306 square miles.
237. It is by this rule that the measure in (215) is deduced from that in (214); for it is evident that twelve inches being a foot, the square foot is 12 × 12 or 144 square inches, and so on. In a similar way it may be shewn that the content in cubic inches of a cube, or parallelepiped,[48] may be found by multiplying together the number of inches in those three sides which meet in a point. Thus, a cube of 6 inches contains 6 × 6 × 6, or 216 cubic inches; a chest whose sides are 6, 8, and 5 feet, contains 6 × 8 × 5, or 240 cubic feet. By this rule the measure in (216) was deduced from that in (214).
SECTION II.
RULE OF THREE.
238. Suppose it required to find what 156 yards will cost, if 22 yards cost 17s. 4d. This quantity, reduced to pence, is 208d.; and if 22 yards cost 208d., each yard costs ²⁰⁸/₂₂d. But 156 yards cost 156 times the price of one yard, and therefore cost
| 208 | × 156 pence, or | 208 × 156 | pence (117). |
| 22 | 22 |
Again, if 25½ French francs be 20 shillings sterling, how many francs are in £20. 15? Since 25½ francs are 20 shillings, twice the number of francs must be twice the number of shillings; that is, 51 francs are 40 shillings, and one shilling is the fortieth part of 51 francs, or ⁵¹/₄₀ francs. But £20 15s. contain 415 shillings (219); and since 1 shilling is ⁵¹/₄₀ francs, 415 shillings is